Outer products

So long as the field is on, these populations continue to change however, once the external field is turned off, these populations remain constant (discounting relaxation processes, which will be introduced below). Yet the amplitudes in the states i and i / do continue to change with time, due to the accumulation of time-dependent phase factors during the field-free evolution. We can obtain a convenient separation of the time-dependent and the time-mdependent quantities by defining a density matrix, p. For the case of the wavefiinction ), p is given as the outer product of v i) with itself. 

This outer product gives four temis, which may be arranged in matrix fonn as... 

Dyadic notation is used in this equation, so X.i- denotes the outer product... 

Fig. 29.6. Schematic illustration of four types of special matrix products the matrix-by-vector product, the vector-by-matrix product, the outer product and the scalar product between vectors, respectively from top to bottom.

The outer product results from premultiplying an n vector x with a p vector... 

The outer product of two vectors can be thought of as the matrix product between a single-column matrix with a single-row matrix ... 

Multiplication of Am by Bn yields a mixed tensor AmBn = C , called the outer product, which may be formed with tensors of any rank or type,... 

The basis for the analysis using the SCM is illustrated in Figure 9.3. The gas film, outer product (ash) layer, and unreacted core of B are three distinct regions. We derive the continuity equation for A by means of a material balance across a thin spherical shell in the ash layer at radial position r and with a thickness dr. The procedure is the same as that leading up to equation 9.1-5, except that there is no reaction term involving (- rA), since no reaction occurs in the ash layer. The result corresponding to equation 9.1-5 is... 

The outer product of two vectors xm and y is the matrix Amx , such that... 

Let us prove a useful alternative expression of the matrix product. Let e,(i = l, n) be the column vector whose n coordinates are zero except for the ith which is equal to 1. The n e-s form a base of the Euclidian space 91". Ue, is the ith column of a matrix Umxn while ef V is the ith row of a matrix V x p. Outer products such as etef are n x n matrices. From the previous definitions... 

The common-dimension expansion shows that the matrix product can be viewed as a linear combination of the pairwise outer products of U columns and V rows. Example ... 

No bold face will be used for the subscript X of T,x. The current element (il, i2) of can be rewritten as S(Xn Xi2)—S(Xil)S(Xi2). The condensed form of the covariance-matrix is obtained by using the outer product defined in Section 2.1... 

The product of a column vector with m rows and a row vector with n columns results in a matrix with m rows and n columns. This is the so-called outer product. 

The matrix product (outer product) of two vectors is a matrix (example abT. The first vector must be a column vector, and the second a row vector (see Figure A.2.3). 

This construction in which a vector is used to form a matrix v(i)Xv(i) is called an "outer product". The projection matrix thus formed can be shown to be idempotent, which means that the result of applying it twice (or more times) is identical to the result of applying it once P P = P. This property is straightforward to demonstrate. Let us consider... 

Equation (9.15) defines the scalar product (or inner product, dot product vjv7), and (9.16) defines the corresponding dyadic product (or outer product y7yl) of vectors v -and Vj, which are different kinds of mathematical object. 

Just as a reminder The dots between the vectors denote the scalar (inner) product and the crosses denote the cross (outer) product of the vectors. These vectors 6 are in units of nr, which is proportional to the inverse of the lattice constants of the real space crystal lattice. This is why one calls the three-dimensional space spanned by these vectors the reciprocal space and the lattice defined by these primitive vectors is called the reciprocal lattice. These primitive reciprocal vectors have the following properties ... 

In Equation 4.2, the product c,ptT is the outer product of the concentration profile of the first pure component times its pure spectrum. 

There is still debate over the proper manner to calculate the NAS. In the earliest work by Ho et al. [20], the three-way NAS is calculated as the outer product of the multivariate NAS from the resolved X-way and Y-way profiles, such that... 

Similarly, Messick et al. [44] suggested that the NAS can be found by orthogonal projection of Equation 12.16 following unfolding each / x J sample and interferent matrix into an /./ x I vector. The three-way NAS is the consequent NAS of Equation 12.16 refolded into an I x J matrix. The third alternative, propounded by Wang et al. [41], is to construct the NAS from the outer products of the X-way and Y-way profiles that are unique to the analyte. In this method, no projections are explicitly calculated. 

Figure 8. Pictorial representation of the outer product matrix multiplication algorithm. ...

Fortran/VPLIB Code for Matrix Matrix "Outer Product"... 

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